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In this article, we give an overview of scale-spaces and their application to noise suppression and segmentation of 1-D signals and 2-D images. Several prototypical problems serve as our motivation. We review several scale-spaces (linear Gaussian, Perona-Malik, and SIDE-stabilized inverse diffusion equation) and discuss their advantages and shortcomings. We describe our previous work and argue that a very simple nonlinear scale-space leads to a fast estimation algorithm which produces accurate segmentations and estimates of signals and images.